- 2016 - date: PhD student, DAMTP, CCA, University of Cambridge.
- 2015 - 2016: M.Math. (Part III), University of Cambridge.
- 2012 - 2015: B.A. Maths, University of Cambridge.
Research
Matthew is a member of the Applied Functional and Harmonic Analysis group within the Cambridge Centre for Analysis at DAMTP. Details of his research can be found here. His current interests are:
- Spectral theory, in particular the existence (or non-existence!) of algorithms to solve computational problems.
- Deep learning, its mathematical foundations and the problem of stability.
- Spectral/pseudospectral methods and linear algebra in infinite dimensions.
- Scattering problems and other problems related to computational fluid mechanics.
Publications
A full list of Matthew's publications can be seen here. Below is a list updated September 2019.
Journal Articles
(*denotes first authors where applicable, others alphabetical)
(paper prizes shown in orange)
Computational Spectral Theory
| Pseudoergodic operators and periodic boundary conditions Matthew Colbrook* Mathematics of Computation, 2019. preprint journal link |
| How to compute spectra with error control (Smith-Knight/Rayleigh-Knight Prize Class I 2018) Matthew Colbrook*, Bogdan Roman, Anders Hansen Physical Review Letters, 2019 (chosen as cover article) preprint journal link |
| On the infinite-dimensional QR algorithm Matthew Colbrook*, Anders Hansen Numerische Mathematik, 2019. preprint journal link |
Spectral Methods
| A hybrid analytical-numerical technique for elliptic PDEs Matthew Colbrook*, Thanasis Fokas, Parham Hashemzadeh SIAM Journal on Scientific Computing (SISC), 2019. preprint journal link |
| Extending the unified transform: curvilinear polygons and variable coefficient PDEs Matthew Colbrook* IMA Journal of Numerical Analysis, 2018. preprint journal link |
| On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains Matthew Colbrook*, Natasha Flyer, Bengt Fornberg Journal of Computational Physics, 2018. preprint journal link |
Scattering Problems and Computational Fluid Mechanics
| A spectral collocation method for acoustic scattering by multiple elastic plates Matthew Colbrook*, Lorna Ayton Journal of Sound and Vibration, 2019. preprint journal link |
| The unified transform for mixed boundary condition problems in unbounded domains (IMA Lighthill-Thwaites 2nd Prize 2019) Matthew Colbrook*, Lorna Ayton, Athanassios Fokas Proceedings of the Royal Society A, 2019. preprint journal link |
| Scaling laws of passive-scalar diffusion in the interstellar medium Matthew Colbrook*, Xiangcheng Ma, Philip Hopkins, Jonathan Squire Monthly Notices of the Royal Astronomical Society, 2017. preprint journal link |
Other
| A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line Felipe de Barros, Matthew Colbrook, Athanassios Fokas International Journal of Heat and Mass Transfer, 2019. preprint journal link |
| Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons Matthew Colbrook*, Athanassios Fokas Applied Numerical Mathematics, 2018. preprint journal link |
Refereed Conference Articles
| The unified transform: a spectral collocation method for acoustic scattering Lorna Ayton, Matthew Colbrook, Athanassios Fokas AIAA/CEAS Aeroacoustics, 2019. preprint journal link |
| On the existence of stable and accurate neural networks for image reconstruction Matthew Colbrook*, Vegard Antun, Anders Hansen Signal Processing with Adaptive Sparse Structured Representations (SPARS), 2019. preprint |
Submitted Articles/Preprints
| Computing spectral measures and spectral types: new algorithms and classifications Matthew Colbrook* arXiv preprint |
| The foundations of spectral computations via the solvability complexity index hierarchy: Part I Matthew Colbrook*, Anders Hansen arXiv preprint |
| The foundations of spectral computations via the solvability complexity index hierarchy: Part II Matthew Colbrook* arXiv preprint |
| On the solvability complexity index hierarchy and towers of algorithms Jonathan Ben-Artzi, Matthew Colbrook, Anders Hansen, Olavi Nevanlinna, Markus Seidel arXiv preprint |
| Kernel density estimation with linked boundary conditions Matthew Colbrook*, Zdravko Botev, Karsten Kuritz, Shev MacNamara arXiv preprint |
| Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions) Matthew Colbrook, Matthew Priddin preprint |
| Stable and accurate neural networks for image reconstruction Matthew Colbrook* In prep. |
Publications
On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains
– Journal of Computational Physics
(2018)
374,
996
(DOI: 10.1016/j.jcp.2018.08.005)
Extending the unified transform: curvilinear polygons and variable coefficient PDEs
– IMA Journal of Numerical Analysis
(2018)
(DOI: 10.1093/imanum/dry085)
Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons
– Applied Numerical Mathematics
(2018)
126,
1
(DOI: 10.1016/j.apnum.2017.12.001)
Scaling laws of passive-scalar diffusion in the interstellar medium
– Monthly Notices of the Royal Astronomical Society
(2017)
467,
2421
(DOI: 10.1093/mnras/stx261)
Kernel Density Estimation with Linked Boundary Conditions